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  "The ",
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  "Journal"
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Cell[TextData[{
  StyleBox[" Support Vector Classifier  via ",
    FontFamily->"Times New Roman"],
  StyleBox["Mathematica",
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Cell["\<\

 B. Pal\[AAcute]ncz
 
 Department of Photogrammetry and Geoinformatics
 Budapest University of Technology  and Economics, 
 H-1111 Budapest, Hungary, M\[UDoubleDot]egyetem rkp. 3.
 
 e-mail:  palancz@epito.bme.hu\
\>", "Author",
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Cell[CellGroupData[{

Cell["Abstract", "Section"],

Cell[TextData[{
  "In this case study, a  Support Vector Classifier function has been \
developed in ",
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  " Starting with a brief summary of support vector classification method, \
the step by step implementation of the classification algorithm in  ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  "  is presented and explained. To check our function, two test problems, \
learning a chess board and classification of two intertwined spirals are \
solved. In addition an application to image recognition is demonstrated using \
wavelet transformation as feature extraction method."
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  "1. Introductio",
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Cell["\<\
Kernel Methods are relatively new family of algorithms that presents a series \
of useful features for pattern analysis in datasets. KMs combine the \
simplicity and computational efficiency of linear algorithms, such as the \
perception algorithm or ridge regression, with flexibility of nonlinear \
systems, such as for example neural networks, and rigor of statistical \
approaches such as regularization methods in multivariate statistics. As a \
result of the special way they represent functions, these algorithms \
typically reduce the learning step to convex optimization problem, that can \
always be solved in polynomial time, avoiding the problem of local minima \
typical of neural networks, decision trees and other nonlinear approaches.\
\>", "Text"]
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Cell[TextData[{
  "2. Support Vector Classificatio",
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Cell["2.1 Binary classification", "Subsection"],

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  "\tIn case of binary classification, we try to estimate a real-valued \
function  ",
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",
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that we choose our estimate ",
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example by satisfying ",
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additional about ",
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patterns carry no information whatsoever about values on novel patterns. \
Hence learning is impossible, and minimizing the training error does not \
imply a small expected test error.\n\tStatistical learning theory, or VC \
(Vapnik-Chervonenkis) theory, shows that is crucial to restrict the class of \
functions that the learning machine can implement to one with capacity that \
is suitable for the amount of available training data.\n\t"
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Cell["2.2 Optimal hyperplane classifier", "Subsection"],

Cell["\<\
To design learning algorithms, we thus must come up with a class of  \
functions whose capacity can be computed. SV classifiers are based on the \
class of hyperplanes\
\>", "Text"],

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We can show that the optimal hyperplane, defined as the one with the maximal \
margin of separation between the two classes (see Figure 1), has the lowest \
capacity, which ensuring that the classifier learned from training samples \
will misclassify the less elements of the test samples originated from the \
same probability distribution.\
\>", "Text"],

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  "Figure.1  A separable classification problem. The optimal hyperplane is \
orthogonal to the shortest line connecting the convex hulls of the two \
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Cell["2.3 Maximal margin  classifier", "Subsection"],

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  "The traing patterns lie closest to the hyperplane  (see Figure 1. two red \
balls and one blue diamond) are called support vectors, carrying all relevant \
information about the classification problem. The number of support vectors, \
",
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Cell["2.4 Feature spaces and kernels", "Subsection"],

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  "Figure 2. The idea of SV machines: map the training data nonlinearly into \
a higher dimensional feature space via ",
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  "and construct a separating hyperplane with maximum margin there. This \
yields a nonlinear decision boundary in input space. By the use of kernel \
function, it is possible to compute the separating hyperplane without \
explicitly carrying out the map into the feature space."
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  "\nClearly, if ",
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  " is high dimensional, the dot product on the righthand side will be very \
expensive to compute. In some cases, however there is a simple kernel that \
can be evaluated efficiently. For instance, the polynomial kernel "
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  "can be shown to correspond to a map",
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Cell["\<\
3.1 Steps of implementation 
\
\>", "Subsubsection",
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for the output, is \n"
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Let us employ Gaussian kernel with \[Beta] gain  \
\>", "Text"],

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      0.8333963921313136`\ \[ExponentialE]\^\(\(-10.`\)\ \((x\^2 + \((\(-1\) \
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      0.8333963921313133`\ \[ExponentialE]\^\(\(-10.`\)\ \((\((\(-1\) + \
x)\)\^2 + y\^2)\)\) - 
      0.8333963921313133`\ \[ExponentialE]\^\(\(-10.`\)\ \((x\^2 + \
y\^2)\)\)\)], "Output",
  CellLabel->"Out[26]="]
}, Open  ]],

Cell["\<\
Let us display the contour lines of the continuous classification function,\
\>", "Text"],

Cell[BoxData[
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  CellLabel->"In[27]:="],

Cell["\<\
p=ContourPlot[f[{x,y}], {x, 0, \
1},{y,0,1},Contours->{-0.5,-0.05,0,0.05,0.5},ContourShading\[Rule]False,\
PlotPoints\[Rule]50,DisplayFunction\[Rule]Identity];\
\>", "Input",
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  CellTags->"S5.34.1"],

Cell[BoxData[
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Cell["<<Graphics`MultipleListPlot`", "Input",
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Cell[BoxData[
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Cell[CellGroupData[{

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  StyleBox["Figure 3",
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  ". The contour lines of  the continuous classification function ",
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  "(",
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  "1, ",
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  "2) for XOR problem"
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The discrete classifier, the decision rule using signum function is\
\>", "Text"],

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Cell[TextData[{
  StyleBox["Figure 4. ",
    FontVariations->{"CompatibilityType"->0}],
  "The decision rule,  sign(",
  StyleBox["f",
    FontSlant->"Italic"],
  "(",
  StyleBox["x",
    FontSlant->"Italic"],
  "1, ",
  StyleBox["x",
    FontSlant->"Italic"],
  "2)) for XOR problem"
}], "Caption",
  TextAlignment->Center],

Cell["Let us check the classifier", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Map[Sign[f[#]] &, xym]\)], "Input",
  CellLabel->"In[34]:="],

Cell[BoxData[
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  CellLabel->"Out[34]="]
}, Open  ]],

Cell["and", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[35]:="],

Cell[BoxData[
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}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[{
  "3.2 ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " module for SVR"
}], "Subsubsection",
  CellDingbat->None,
  FontSize->14],

Cell[TextData[{
  "These steps can be collected in a module, where the vector ",
  StyleBox["xm",
    FontSlant->"Italic"],
  " contains the input vectors (the training set), and the  vector ",
  StyleBox["ym ",
    FontSlant->"Italic"],
  StyleBox["contains the corresponding scalar output values ( the labels of \
the training set),",
    FontVariations->{"CompatibilityType"->0}]
}], "Text"],

Cell[BoxData[
    \(\(SupportVectorClassifier[xm_, ym_, K_, c_] := 
        Module[\[IndentingNewLine]{m, n, M, i, j, W, g, vars, sol, bdata, 
            b}, \[IndentingNewLine]m = Length[ym]; 
          n = Length[xm[\([1]\)]]; \[IndentingNewLine]M = 
            Table[K[xm[\([i]\)], xm[\([j]\)]], {i, 1, m}, {j, 1, 
                  m}] + \(1\/c\) 
                IdentityMatrix[
                  m]; \[IndentingNewLine]W = \[Sum]\+\(i = 1\)\%m \[Alpha]\_i \
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ym[\([j]\)]\ \(\[Alpha]\_i\) \[Alpha]\_j\ M[\([i, 
                            j]\)])\)\)\); \[IndentingNewLine]g = 
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                          ym[\([i]\)]\ \[Alpha]\_i \[Equal] 
                    0}]]; \[IndentingNewLine]vars = 
            Table[\[Alpha]\_i, {i, 1, 
                m}]; \[IndentingNewLine]sol = \(NMaximize[{W, g}, vars, 
                Method \[Rule] 
                  DifferentialEvolution]\)[\([2]\)]; \[IndentingNewLine]bdata \
= Table[\((\(1 - \[Alpha]\_j\/c\)\/ym[\([j]\)] - \[Sum]\+\(i = 1\)\%m 
                          ym[\([i]\)]\ \[Alpha]\_i\ K[xm[\([i]\)], 
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                          xm[\([i]\)]])\) + b)\) /. sol, 
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  CellLabel->"In[2]:="],

Cell[TextData[{
  "The results of this module are the analytic form of the continuous \
classifier function and the values of  ",
  Cell[BoxData[
      \(\[Alpha]\_i\)]],
  "'s. Let us check the solution of the XOR problem"
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[29]:="],

Cell[BoxData[
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x\_1)\)\^2 + \((\(-1\) + x\_2)\)\^2)\)\) + 
        0.8333963921313136`\ \[ExponentialE]\^\(\(-10.`\)\ \((x\_1\%2 + \
\((\(-1\) + x\_2)\)\^2)\)\) + 
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x\_1)\)\^2 + x\_2\%2)\)\) - 
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell["4. Test problems", "Section"],

Cell[CellGroupData[{

Cell["4.1 Learning a chess board ", "Subsubsection",
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  FontSize->14],

Cell["\<\
Let us consider a 2 \[Times] 2 chess board. The training points are generated \
by uniformly distributed random numbers from the interval [-1,1] \[Times] \
[-1, 1]. 
The chess board matrix\
\>", "Text"],

Cell[BoxData[
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Cell[BoxData[
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  CellLabel->"In[38]:="],

Cell["Creating the training set using 50 random samples,", "Text"],

Cell[BoxData[
    \(xym = {}; zm = {};\)], "Input",
  CellLabel->"In[50]:="],

Cell[BoxData[
    \(\(Do[x1 = Random[Real, {\(-0.99\), 0.99}]; 
        x2 = Random[Real, {\(-0.99\), 0.99}]; \[IndentingNewLine]If[
          x1\ x2 > 0, z = 1, z = \(-1\)]; AppendTo[xym, {x1, x2}]; 
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Cell["Preparation data to display them", "Text"],

Cell[BoxData[
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        Select[data, #[\([3]\)] < 0 &]];\)], "Input",
  CellLabel->"In[53]:="],

Cell["\<\
p2=MultipleListPlot[data1,data2,SymbolShape -> {PlotSymbol[Triangle,5], \
PlotSymbol[Box,2]},SymbolStyle\[Rule]{Hue[.7],Hue[.0]},Frame\[Rule]True,\
AspectRatio\[Rule]1,DisplayFunction\[Rule]Identity];\
\>", "Input",
  CellLabel->"In[54]:=",
  CellTags->"S5.37.1"],

Cell["\<\
Let us employ the same Gaussian kernel , but now with gain  \[Beta] = 15.  \
\>", "Text"],

Cell[BoxData[
    \(\(\[Beta] = 15;\)\)], "Input",
  CellLabel->"In[44]:="],

Cell[TextData[{
  "employing parameter ",
  StyleBox["c",
    FontSlant->"Italic"],
  " = 100,"
}], "Text"],

Cell[BoxData[
    \(\(c = 100;\)\)], "Input",
  CellLabel->"In[45]:="],

Cell["The solution is", "Text"],

Cell[BoxData[
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Cell["This run could take some minutes.", "Text"],

Cell[TextData[{
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  ",-1,1},{",
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DisplayFunction\[Rule]Identity];"
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Cell[TextData[{
  "Figure 5. SVC result from 60 random samples: the functions  f(x) with the \
random samples, the ideal chess board and sign(",
  StyleBox["f",
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  "(",
  StyleBox["x",
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}], "Caption",
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Cell["\<\
The SV solution of a larger chess board problem can be found in [3].\
\>", "Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell["4.2 Two intertwined spirals  ", "Subsubsection",
  CellDingbat->None,
  FontSize->14],

Cell["\<\
The two intertwined spirals is a challenging classification benchmark \
originated from the field of neural networks [4].\
\>", "Text"],

Cell["Parametric equations of the spirals are", "Text"],

Cell[BoxData[
    \(x1[t_] := 2\ Cos[\ t]\ \[ExponentialE]\^\(t/10\)\)], "Input",
  CellLabel->"In[2]:="],

Cell[BoxData[
    \(y1[t_] := 1.5\ Sin[\ t]\ \[ExponentialE]\^\(t/10\)\)], "Input",
  CellLabel->"In[3]:="],

Cell[BoxData[
    \(x2[t_] := 2.7  Cos[\ t]\ \[ExponentialE]\^\(t/10\)\)], "Input",
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Cell[BoxData[
    \(y2[t_] := 2.025  Sin[\ t]\ \[ExponentialE]\^\(t/10\)\)], "Input",
  CellLabel->"In[5]:="],

Cell["Generating 26 discrete points for each spiral,", "Text"],

Cell[BoxData[
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  CellLabel->"In[24]:="],

Cell[BoxData[
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  CellLabel->"In[25]:="],

Cell["then displaying these points,", "Text"],

Cell[BoxData[
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          DisplayFunction \[Rule] Identity];\)\)], "Input",
  CellLabel->"In[26]:="],

Cell[BoxData[
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          DisplayFunction \[Rule] Identity];\)\)], "Input",
  CellLabel->"In[27]:="],

Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"From In[28]:=",
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}, Open  ]],

Cell["\<\
Figure 6. Two intertwined spirals represented by 41 points each\
\>", "Caption",
  TextAlignment->Center],

Cell["\<\
Creating the teaching set, putting these points into one list\
\>", "Text"],

Cell[BoxData[
    \(\(xym = Join[s1, s2];\)\)], "Input",
  CellLabel->"In[29]:="],

Cell["Generating the labels of the samples,", "Text"],

Cell[BoxData[
    \(\(zm = Join[Table[1. , {26}], Table[\(-1. \), {26}]];\)\)], "Input",
  CellLabel->"In[33]:="],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[35]:="],

Cell[BoxData[
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Cell[TextData[{
  "Applying  wavelet kernel [5] with parameter ",
  StyleBox["a",
    FontSlant->"Italic"],
  " = 1.8,  in case of  dimension  ",
  StyleBox["n",
    FontSlant->"Italic"],
  " = 2 "
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Cell[BoxData[
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  CellLabel->"In[36]:="],

Cell[BoxData[
    \(K[u_, 
        v_] := \[Product]\+\(i = 1\)\%n\((Cos[
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\([i]\)] - v[\([i]\)])\)\^2\)\)\/\(2\ a\^2\)\)\)])\)\)], "Input",
  CellLabel->"In[37]:="],

Cell[TextData[{
  "and with parameter ",
  StyleBox["c",
    FontSlant->"Italic"],
  " = 100"
}], "Text"],

Cell[BoxData[
    \(\(c = 100. ;\)\)], "Input",
  CellLabel->"In[38]:="],

Cell["The solution is", "Text"],

Cell[BoxData[
    \(\(F = SupportVectorClassifier[xym, zm, K, c];\)\)], "Input",
  CellLabel->"In[39]:="],

Cell["This run could take some minutes.", "Text"],

Cell[TextData[{
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  "ContourShading\[Rule]False,DisplayFunction\[Rule]Identity];"
}], "Input",
  CellLabel->"In[40]:=",
  CellTags->"S5.34.1"],

Cell[CellGroupData[{

Cell[BoxData[
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Figure 7. Classification results of SVC with the nonlinear decision \
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The continuous classification function and the decision rule can be displayed \
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Cell["\<\
Figure 8. Classification function and the corresponding decision rule\
\>", "Caption",
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Cell[CellGroupData[{

Cell["5. Image classification", "Section"],

Cell["\<\
We should like to classify  images into two different categories, snowman and \
dice, see Figures 9 and 10.
\
\>", "Text"],

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  "The feature extraction of these pictures  represented by  matrices of 128 \
\[Times] 128 pixels,  was carried out by wavelet transform using second order \
Daubechies filter [6], then employing averaging technique [ 7] for seven \
nonoverlapping bands of the spectrum. Consequently the dimension of these \
feature vectors is seven,  ",
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The first 7 -7 pictures are considered as teaching set and the last 3 -3 \
pictures are considered as test samples.\
\>", "Text"],

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Cell["\<\
Let us labeling the snowmen category with  +1  and the dice category with   - \
1.\
\>", "Text"],

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Cell["References", "Section"],

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  "\n",
  "1. Berthold M. and Hand D.J. (Eds.) \"Intelligent Data Analysis, An \
Introduction\", Springer, 2003.\n \n2. Hearst M. A. ",
  StyleBox["Support Vector Machines,",
    FontSlant->"Italic"],
  " IEEE Intelligent Systems, pp. 18 -28, July/August, 1998.\n\n3. \
Cristianini N. and Shawe-Taylor J. \"An introduction to Support Vector \
Machines and other kernel - based learning methods\", Cambridge, University \
Press, 2003.",
  "\n\n4.Juill\[EAcute] H.,Pollack J.B. ",
  StyleBox["Co-evolving intertwined spirals",
    FontSlant->"Italic"],
  ", in Proc. of the 5th. Ann.Conf. on Evolutionary Programming, San Diego, \
USA, pp.461-468, 1996.\n\n5. ",
  "Zhang L, Zhou W. and Jiao L. ",
  StyleBox["Wavelet Support Vector Machine",
    FontSlant->"Italic"],
  ", IEEE Trans. Systems, Man and Cybernetics - Part B: Cybernetics, Vol. 4. \
No.1. pp. 34 -39, Febr. 2004  ",
  "\n\n6. Wavelet Explorer with ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  ", ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " Application Package, Wolfram Research Inc. 2003.\n\n7. Goswani J.C. and \
Chan A.K. \"Fundamentals of wavelets\", J.Wiley, 1999."
}], "Reference"]
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